For many materials,
linear elastic models do not accurately describe the observed material behaviour. The most common example of this kind of material is rubber, whose
stress-
strain relationship can be defined as non-linearly elastic,
isotropic and
incompressible. Hyperelasticity provides a means of modeling the stress–strain behavior of such materials.[2] The behavior of unfilled,
vulcanizedelastomers often conforms closely to the hyperelastic ideal. Filled elastomers and
biological tissues[3][4] are also often modeled via the hyperelastic idealization.
The simplest hyperelastic material model is the Saint Venant–Kirchhoff model which is just an extension of the geometrically linear elastic material model to the geometrically nonlinear regime. This model has the general form and the isotropic form respectively
where is tensor contraction, is the second Piola–Kirchhoff stress, is a fourth order
stiffness tensor and is the Lagrangian Green strain given by
and are the
Lamé constants, and is the second order unit tensor.
The strain-energy density function for the Saint Venant–Kirchhoff model is
and the second Piola–Kirchhoff stress can be derived from the relation
Classification of hyperelastic material models
Hyperelastic material models can be classified as:
Generally, a hyperelastic model should satisfy the
Drucker stability criterion.
Some hyperelastic models satisfy the
Valanis-Landel hypothesis which states that the strain energy function can be separated into the sum of separate functions of the
principal stretches:
The above expressions are valid even for anisotropic media (in which case, the potential function is understood to depend implicitly on reference directional quantities such as initial fiber orientations). In the special case of isotropy, the Cauchy stress can be expressed in terms of the left Cauchy-Green deformation tensor as follows:[7]
Incompressible hyperelastic materials
For an
incompressible material . The incompressibility constraint is therefore . To ensure incompressibility of a hyperelastic material, the strain-energy function can be written in form:
where the hydrostatic pressure functions as a
Lagrangian multiplier to enforce the incompressibility constraint. The 1st Piola–Kirchhoff stress now becomes
This stress tensor can subsequently be
converted into any of the other conventional stress tensors, such as the
Cauchy stress tensor which is given by
where is an undetermined pressure which acts as a
Lagrange multiplier to enforce the incompressibility constraint.
If, in addition, , we have and hence
In that case the Cauchy stress can be expressed as
Proof 2
The
isochoric deformation gradient is defined as , resulting in the isochoric deformation gradient having a determinant of 1, in other words it is volume stretch free. Using this one can subsequently define the isochoric left Cauchy–Green deformation tensor .
The invariants of are
The set of invariants which are used to define the distortional behavior are the first two invariants of the isochoric left Cauchy–Green deformation tensor tensor, (which are identical to the ones for the right Cauchy Green stretch tensor), and add into the fray to describe the volumetric behaviour.
To express the Cauchy stress in terms of the invariants recall that
The chain rule of differentiation gives us
Recall that the Cauchy stress is given by
In terms of the invariants we have
Plugging in the expressions for the derivatives of in terms of , we have
or,
In terms of the deviatoric part of , we can write
For an
incompressible material and hence .Then
the Cauchy stress is given by
where is an undetermined pressure-like Lagrange multiplier term. In addition, if , we have and hence
the Cauchy stress can be expressed as
Proof 3
To express the Cauchy stress in terms of the
stretches recall that
The chain rule gives
The Cauchy stress is given by
Plugging in the expression for the derivative of leads to
where is an undetermined pressure. In terms of stress differences
If in addition , then
If , then
Consistency with linear elasticity
Consistency with linear elasticity is often used to determine some of the parameters of hyperelastic material models. These consistency conditions can be found by comparing
Hooke's law with linearized hyperelasticity at small strains.
Consistency conditions for isotropic hyperelastic models
For isotropic hyperelastic materials to be consistent with isotropic
linear elasticity, the stress–strain relation should have the following form in the
infinitesimal strain limit:
where are the
Lamé constants. The strain energy density function that corresponds to the above relation is[1]
For an incompressible material and we have
For any strain energy density function to reduce to the above forms for small strains the following conditions have to be met[1]
If the material is incompressible, then the above conditions may be expressed in the following form.
These conditions can be used to find relations between the parameters of a given hyperelastic model and shear and bulk moduli.
Consistency conditions for incompressible I1 based rubber materials
Many elastomers are modeled adequately by a strain energy density function that depends only on . For such materials we have .
The consistency conditions for incompressible materials for may then be expressed as
The second consistency condition above can be derived by noting that
These relations can then be substituted into the consistency condition for isotropic incompressible hyperelastic materials.
For many materials,
linear elastic models do not accurately describe the observed material behaviour. The most common example of this kind of material is rubber, whose
stress-
strain relationship can be defined as non-linearly elastic,
isotropic and
incompressible. Hyperelasticity provides a means of modeling the stress–strain behavior of such materials.[2] The behavior of unfilled,
vulcanizedelastomers often conforms closely to the hyperelastic ideal. Filled elastomers and
biological tissues[3][4] are also often modeled via the hyperelastic idealization.
The simplest hyperelastic material model is the Saint Venant–Kirchhoff model which is just an extension of the geometrically linear elastic material model to the geometrically nonlinear regime. This model has the general form and the isotropic form respectively
where is tensor contraction, is the second Piola–Kirchhoff stress, is a fourth order
stiffness tensor and is the Lagrangian Green strain given by
and are the
Lamé constants, and is the second order unit tensor.
The strain-energy density function for the Saint Venant–Kirchhoff model is
and the second Piola–Kirchhoff stress can be derived from the relation
Classification of hyperelastic material models
Hyperelastic material models can be classified as:
Generally, a hyperelastic model should satisfy the
Drucker stability criterion.
Some hyperelastic models satisfy the
Valanis-Landel hypothesis which states that the strain energy function can be separated into the sum of separate functions of the
principal stretches:
The above expressions are valid even for anisotropic media (in which case, the potential function is understood to depend implicitly on reference directional quantities such as initial fiber orientations). In the special case of isotropy, the Cauchy stress can be expressed in terms of the left Cauchy-Green deformation tensor as follows:[7]
Incompressible hyperelastic materials
For an
incompressible material . The incompressibility constraint is therefore . To ensure incompressibility of a hyperelastic material, the strain-energy function can be written in form:
where the hydrostatic pressure functions as a
Lagrangian multiplier to enforce the incompressibility constraint. The 1st Piola–Kirchhoff stress now becomes
This stress tensor can subsequently be
converted into any of the other conventional stress tensors, such as the
Cauchy stress tensor which is given by
where is an undetermined pressure which acts as a
Lagrange multiplier to enforce the incompressibility constraint.
If, in addition, , we have and hence
In that case the Cauchy stress can be expressed as
Proof 2
The
isochoric deformation gradient is defined as , resulting in the isochoric deformation gradient having a determinant of 1, in other words it is volume stretch free. Using this one can subsequently define the isochoric left Cauchy–Green deformation tensor .
The invariants of are
The set of invariants which are used to define the distortional behavior are the first two invariants of the isochoric left Cauchy–Green deformation tensor tensor, (which are identical to the ones for the right Cauchy Green stretch tensor), and add into the fray to describe the volumetric behaviour.
To express the Cauchy stress in terms of the invariants recall that
The chain rule of differentiation gives us
Recall that the Cauchy stress is given by
In terms of the invariants we have
Plugging in the expressions for the derivatives of in terms of , we have
or,
In terms of the deviatoric part of , we can write
For an
incompressible material and hence .Then
the Cauchy stress is given by
where is an undetermined pressure-like Lagrange multiplier term. In addition, if , we have and hence
the Cauchy stress can be expressed as
Proof 3
To express the Cauchy stress in terms of the
stretches recall that
The chain rule gives
The Cauchy stress is given by
Plugging in the expression for the derivative of leads to
where is an undetermined pressure. In terms of stress differences
If in addition , then
If , then
Consistency with linear elasticity
Consistency with linear elasticity is often used to determine some of the parameters of hyperelastic material models. These consistency conditions can be found by comparing
Hooke's law with linearized hyperelasticity at small strains.
Consistency conditions for isotropic hyperelastic models
For isotropic hyperelastic materials to be consistent with isotropic
linear elasticity, the stress–strain relation should have the following form in the
infinitesimal strain limit:
where are the
Lamé constants. The strain energy density function that corresponds to the above relation is[1]
For an incompressible material and we have
For any strain energy density function to reduce to the above forms for small strains the following conditions have to be met[1]
If the material is incompressible, then the above conditions may be expressed in the following form.
These conditions can be used to find relations between the parameters of a given hyperelastic model and shear and bulk moduli.
Consistency conditions for incompressible I1 based rubber materials
Many elastomers are modeled adequately by a strain energy density function that depends only on . For such materials we have .
The consistency conditions for incompressible materials for may then be expressed as
The second consistency condition above can be derived by noting that
These relations can then be substituted into the consistency condition for isotropic incompressible hyperelastic materials.